Tip: For best texture, make sure to remove the tough core from the Brussels sprouts (I cut them in half and then slice a V-shape at the base) and then slice them very thin. For quicker shredding, you may add the cored and halved sprouts to a food processor fitted with the thinnest slicing blade. Prep ahead tip: The Brussels sprouts can be shredded up to a day in advance and stored in the fridge in an airtight container or zip-top bag.
For the vinaigrette: Whisk the lemon juice, mustard, shallot, salt, and pepper together in a small bowl or jar with a tight-fitting lid. Slowly whisk in oil until incorporated and the vinaigrette is thick. Or, add the oil, place the lid on the jar, and shake until thick and creamy. Make-ahead tip: This step may be done in advance. The vinaigrette may sit on the counter for up to 2 hours and will keep in the refrigerator for at least a week. Leftovers: You will have leftover vinaigrette for another salad later. Depending on how liberally you dress your salads, it may be enough for another full batch but perhaps a half batch. Or enjoy on other tossed salads (see recipe notes).
Assemble the salad: Place the Brussels sprouts in a large bowl and drizzle with about ⅓ cup of the vinaigrette. Toss, and add more vinaigrette to taste. Then cover and refrigerate for at least 30 minutes or up to 2 hours. (If time is short, letting the salad sit on the counter for even 15 minutes will allow the flavors to meld a bit.)
Fold in the cheese, nuts, and dried fruit. Season with salt and pepper to taste and and serve.